Like linear momentum, angular momentum is fundamentally a vector in
. The definition of the previous section suffices when the
direction does not change, in which case we can focus only on its
magnitude .

More generally, let
denote the 3-space coordinates
of a point-mass, and let
denote its velocity
in . Then the instantaneous angular momentum vector
of the mass relative to the origin (not necessarily rotating about a
fixed axis) is given by

For the special case in which
is orthogonal to
, as in Fig.B.4, we have that
points, by the right-hand rule, in the direction of the angular
velocity vector (up out of the page), which is
satisfying. Furthermore, its magnitude is given by

In the more general case of an arbitrary mass velocity vector
, we know from §B.4.12 that the magnitude of
equals the product of the distance from the axis
of rotation to the mass, i.e.,
, times the length of
the component of
that is orthogonal to
, i.e.,
, as needed.

It can be shown that vector angular momentum, as defined, is
conserved.B.22 For
example, in an orbit, such as that of the moon around the earth, or
that of Halley's comet around the sun, the orbiting object speeds up
as it comes closer to the object it is orbiting. (See Kepler's laws
of planetary motion.) Similarly, a spinning ice-skater spins faster
when pulling in arms to reduce the moment of inertia about the spin
axis. The conservation of angular momentum can be shown to result
from the principle of least action and the isotrophy of space
[270, p. 18].

The vector angular momentum of a rigid body is obtained by summing the
angular momentum of its constituent mass particles. Thus,

Since
factors out of the sum, we see that the mass moment of
inertia tensor for a rigid body is given by the sum of the mass moment
of inertia tensors for each of its component mass particles.

In summary, the angular momentum vector
is given by the mass
moment of inertia tensor
times the angular-velocity vector representing the axis of rotation.

Note that the angular momentum vector
does not in general
point in the same direction as the angular-velocity vector
. We
saw above that it does in the special case of a point mass traveling
orthogonal to its position vector. In general,
and
point
in the same direction whenever
is an eigenvector of
, as will be discussed further below (§B.4.16). In this
case, the rigid body is said to be dynamically balanced.B.24